FW: Middle-School Morphic MathRoger Kenyon edutec at idirect.comSat Jul 28 15:11:35 UTC 2001
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<title>Operations</title>
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word[0]="<b>Consider the facts and try again.</b><hr size='1' noshade><ol><li>Substitute the temperatures given for the variables in the formulas. </li><li>Take the order of operations into account. </li><li>Note that parentheses change the order of operations. </li>";
word[1]="<b>Yes, -40 and 163.</b><hr size='1' noshade><ol><li>Use the strategy of attempt and adjust. <br></li><li>Substitute the temperatures given for the variables in appropriate formula. <br><br>C = (F - 32) * 5 / 9 <br> C = (325 - 32) * 5 / 9 <br>C = (293) * 5 / 9 <br>C = 1465 / 9 <br> C = 162.777 . . .</li>";
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<div align="center"><a name="top"></a><a href="../index.html">Home</a> • <a href="subjects.htm">Subjects</a>
<h2>Fractions</h2>
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<div align="left">After an animal shelter sold one-third of its puppies and gave away half of the remainder, 10 puppies were left. How many puppies were originally in the shelter? Puppies are washed in a large tank. When 24 L of water are poured into the empty tank, the tank becomes 3/4 full. How many liters does the wash tank hold when full?</div>
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<li><a href="#" onClick="popNote(0);return false">30 puppies, 32 L</a></li>
<li><a href="#" onClick="popNote(0);return false">30 puppies, 32 L</a></li>
<li><a href="#" onClick="popNote(0);return false">30 puppies, 32 L</a></li>
<li><a href="#" onClick="popNote(0);return false">30 puppies, 32 L</a></li>
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<div align="center">degrees Celsius</div>
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<p><br>
Upon completion of this strand, the learner should be able to apply to apply arithmetic operations and check accuracy of their outcome, such as using estimation to determine the reasonableness of a multiplication. </p>
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<li>
<div align="left"><a href="#L1">A fraction is a type of fragment</a></div>
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<li>
<div align="left"><a href="#L2">A common fraction compares parts-considered to total-parts</a></div>
</li>
<li>
<div align="left"><a href="#L3">A decimal fraction expresses an amount in place values</a></div>
</li>
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<div align="left"><a href="#L4">A percentage shows parts considered out of 100 equal parts of the whole</a></div>
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<p>Key concepts: fragment.</p>
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<h4><a name="L1"></a><a href="#top">A fraction is a type of fragment.</a> </h4>
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<li> <span onclick="OLine(Sec1)"> <a href="#top">A fraction is an equal share less than a whole</a><a href="#">.</a></span></li>
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<h5>fragment</h5>
<p>“Fraction” and “fragment” both come from the Latin “fractus”, meaning broken.</p>
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<li>
<p>A whole unit (unbroken amount) is an <a href="integers.htm">integer</a>.</p>
</li>
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<p>A fragment is any part of a unit, like a glass fragment broken from a window. A fraction, however is a part of the same size or amount as the other parts.</p>
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<h5>fraction format</h5>
<p>The format of a fraction refers to its expression, not its value.</p>
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<ol>
<li>
<p>A common fraction has the form N/D (such as 3/4) to compare part considered out of total parts.</p>
</li>
<li>
<p>A decimal fraction is expressed in place-values to the right of a decimal point (such as 0.75).</p>
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<li>Percentage is a fractional amount out of 100 equal parts of the whole (such as 75%).</li>
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<h5>aa</h5>
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<p>cc.</p>
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<p>cc.</p>
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<h5>aa</h5>
<p>bb.</p>
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<p>cc.</p>
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<p>cc.</p>
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<br>
<li> <span onclick="OLine(Sec1)"> <a href="#top">A fraction is an equal share less than a whole</a><a href="#">.</a></span></li>
<div class="off" ID="Sec1">
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<h5>fragment</h5>
<p>“Fraction” and “fragment” both come from the Latin “fractus”, meaning broken.</p>
</div>
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<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>A whole unit (unbroken amount) is an <a href="integers.htm">integer</a>.</p>
</li>
<li>
<p>A fragment is any part of a unit, like a glass fragment broken from a window. A fraction, however is a part of the same size or amount as the other parts.</p>
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<h5>fraction format</h5>
<p>The format of a fraction refers to its expression, not its value.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>A common fraction has the form N/D (such as 3/4) to compare part considered out of total parts.</p>
</li>
<li>
<p>A decimal fraction is expressed in place-values to the right of a decimal point (such as 0.75).</p>
</li>
<li>Percentage is a fractional amount out of 100 equal parts of the whole (such as 75%).</li>
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<h5>aa</h5>
<p>bb.</p>
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<p>cc.</p>
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<h5>aa</h5>
<p>bb.</p>
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<li>
<p>cc.</p>
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<p>cc.</p>
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<br>
<li> <span onclick="OLine(Sec2)"> <a href="#L2">dd</a><a href="#">.</a> </span> </li>
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<h5>aa</h5>
<p>bb.</p>
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<li>
<p>cc.</p>
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<p>cc.</p>
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<h5>aa</h5>
<p>bb.</p>
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<p>cc.</p>
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<p>cc.</p>
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<h4><a name="L2"></a><a href="#L2">A common fraction compares parts-considered to total-parts</a><a href="#top">.</a></h4>
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<li> <span onclick="OLine(Sec3)"> <a href="#"> Common fractions can have different forms with the same value.</a></span></li>
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<h5>complex fraction</h5>
<p>In a complex fraction the numerator or the denominator is itself a fraction; e.g. 2/3 / 8.</p>
<p> </p>
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<li>
<p> To simplify a complex fraction, divide the numerator by the denominator.</p>
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<li>
<p>Example: 2/3 / 8 = 2/3 ÷ 1/8 = (2 * 1) / (3 * 8) = 2/24.</p>
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<h5>compound fraction</h5>
<p>A compound fraction is a fraction of a fraction; e.g. 3/4 of 5/9.</p>
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<li>
<p>To simplify a compound fraction, put the product of the numerators over the product of the denominators. </p>
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<li>
<p>Example: 3/8 of 5/6 = 3 * 5 / 8 * 6 = 15/48</p>
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<h5>improper fraction</h5>
<p>In an improper fraction, the numerator is larger than the denominator; e.g. 8/3..</p>
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<li>
<p>To simplify an improper fraction to a mixed number, divide N by D.</p>
</li>
<li>
<p>Example: 22/7 = 3 1/7.</p>
</li>
<li>
<p>Any whole number can be expressed as an improper fraction with 1 as the denominator; e.g. 6 = 6/1.</p>
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<h5>mixed number</h5>
<p>A mixed number is a whole number and a fraction together; e.g. 3 7/8.</p>
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<li>
<p>To change a mixed number into an improper fraction, multiply the whole number by the denominator, add the numerator, place this sum over the denominator.</p>
</li>
<li>
<p>Example: 8 2/5. (8 * 5 + 2) / 5 = 42 / 5.</p>
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<h5>proper fraction</h5>
<p>In a proper fraction, the numerator is smaller than the denominator; e.g. 2/3.</p>
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<li>
<p>cc.</p>
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<li>A proper fraction is called a unit fraction if the numerator is 1, such as 1/72.</li>
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<h5>aa</h5>
<p>bb.</p>
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<li>
<p>cc.</p>
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<p>cc.</p>
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<br>
<li> <span onclick="OLine(Sec1)"> <a href="#top">A fraction is an equal share less than a whole</a><a href="#">.</a></span></li>
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<h5>fragment</h5>
<p>“Fraction” and “fragment” both come from the Latin “fractus”, meaning broken.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>A whole unit (unbroken amount) is an <a href="integers.htm">integer</a>.</p>
</li>
<li>
<p>A fragment is any part of a unit, like a glass fragment broken from a window. A fraction, however is a part of the same size or amount as the other parts.</p>
</li>
</ol>
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<h5>fraction format</h5>
<p>The format of a fraction refers to its expression, not its value.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>A common fraction has the form N/D (such as 3/4) to compare part considered out of total parts.</p>
</li>
<li>
<p>A decimal fraction is expressed in place-values to the right of a decimal point (such as 0.75).</p>
</li>
<li>Percentage is a fractional amount out of 100 equal parts of the whole (such as 75%).</li>
</ol>
</td>
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<td valign="top">
<p>
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<h5>aa</h5>
<p>bb.</p>
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<td width="50%" height="75" bgcolor="#E3EECC">
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<li>
<p>cc.</p>
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<p>cc.</p>
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<h5>aa</h5>
<p>bb.</p>
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</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
<div class="off" ID="Sec3"></div>
<div align="left"><br>
</div>
<li> <span onclick="OLine(Sec4)"> <a href="#"> A common fraction can be changed to equivalent terms.</a></span></li>
<div class="off" ID="Sec4">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>equivalent fractions</h5>
<p>A measurement or ratio can be represented by different but equal fractions.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>3/4, 6/8, 9/12, and 75/100 all have the same value. They reduce to the same value.</p>
</li>
<li>
<p>Two fractions are equivalent if their cross products are equal. 2/3 = 34/51 since 2 * 51 and 3 * 34 both equal 102.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>reduction</h5>
<p>A fraction is reduced to its lowest terms if the terms have no common factors.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>To reduce a fraction, divide N and D with prime numbers until one of the following occurs:
<ol>
<li>N = 1 (e.g.: 1/20), or </li>
<li>D is a prime number (e.g.: 8/17), or</li>
<li>N and D have no common factors (e.g.: 22/85 since 22 = 2 * 11 and 85 = 5 * 17), or</li>
<li>N is a prime number and D is not a multiple of N (e.g.: 7/50)</li>
</ol>
</li>
<li>For example, to reduce 72/90:<br>
<table width="98%" border="0" cellspacing="2" cellpadding="5">
<tr bgcolor="#FFCE9C">
<td width="10%">
<div align="center">step</div>
</td>
<td>
<div align="center">action</div>
</td>
<td width="35%">
<div align="center">result</div>
</td>
</tr>
<tr>
<td width="10%">
<div align="center">1</div>
</td>
<td>
<div align="center">Divide N and D by 2.</div>
</td>
<td width="35%">
<div align="center">72/90 = 36/45</div>
</td>
</tr>
<tr>
<td width="10%">
<div align="center">2</div>
</td>
<td>
<div align="center">45 is not evenly divisibe by 2, so divide N and D by 3.</div>
</td>
<td width="35%">
<div align="center">36/45 = 12/15</div>
</td>
</tr>
<tr>
<td width="10%">
<div align="center">3</div>
</td>
<td>
<div align="center">Again divide N and D by 3.</div>
</td>
<td width="35%">
<div align="center">12/15 = 4/5</div>
</td>
</tr>
<tr>
<td height="27" width="10%">
<div align="center">4</div>
</td>
<td height="27">
<div align="center">The denominator (5) is a prime number, so the algorithm halts.</div>
</td>
<td height="27" width="35%">
<div align="center">72/90 reduces to 4/5</div>
</td>
</tr>
</table>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>monotonic</h5>
<p>The value of a fraction is unchanged if the numerator and denominator are both multiplied or both divided by the same amount.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>1/4 = 1*3/4*3 = 3/12; so 1/4 increases to 3/12.</p>
</li>
<li>
<p> 20/28 = 20÷4/28÷4 = 5/7; so 20/28 reduces to 5/7.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
<p align="center">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>order</h5>
<p>Among fractions with common denominators, that fraction with the largest numerator is overall the largest fraction.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>It may be necessary to first find common denominators.</p>
</li>
<li>
<p>Which is greater, 7/8 or 11/16? Since 7/8 = 14/16, it is greater than 11/16. </p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top" bgcolor="#FFFFFF">
<p align="center">
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<tr>
<td colspan="4" align="center" valign="middle" bordercolor="#FFFFFF" bgcolor="#FFFFFF">
<tr bgcolor="#FFFFFF" bordercolor="#FFFFFF">
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="L3"></a><a href="#top">dd.</a></h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<li> <span onClick="OLine(Sec5)"> <a href="#">ee.</a></span></li>
<div class="off" id="Sec5">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top"> </td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<br>
<li> <span onClick="OLine(Sec6)"> <a href="#">There are various rules of exponent arithmetic.</a> </span> </li>
<div class="off" id="Sec6">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<br>
<li> <span onClick="OLine(Sec7)"> <a href="#">Root finds the factor components of a number.</a> </span> </li>
<div class="off" id="Sec7">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<br>
<li> <span onclick="OLine(Sec8)"> <a href="#">The logarithm of a number is the exponent to which the base must be raised to equal that number.</a></span></li>
<div class="off" ID="Sec8">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<tr bgcolor="#FFFFFF" bordercolor="#FFFFFF">
<td colspan="4" align="center" valign="middle" bordercolor="#FFFFFF" bgcolor="#FFFFFF">
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="L4"></a><a href="#top">Arithmetic operations have properties and precedence.</a> </h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<li> <span onClick="OLine(Sec9)"> <a href="#">Arithmetic properties change an expression's form without changing its value.</a></span></li>
<div class="off" id="Sec9">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top"> </td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top"> </td>
<td width="75%"></td>
</tr>
<td valign="top"> </td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
<td valign="top"> </td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<br>
<li> <span onClick="OLine(Sec10)"> <a href="#">Operations are applied in a conventional order or precedence.</a></span></li>
<div class="off" id="Sec10">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>order of operations</h5>
<p>An operator is a symbol describing the operation to apply to one or more operand. An operand is the data element on which an operation is applied.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li> Hierarchy of frequently used operations:
<table width="95%" border="0" cellspacing="2" cellpadding="5">
<tr bgcolor="#FFCE9C">
<td>
<div align="center">rank</div>
</td>
<td>
<div align="center">symbol</div>
</td>
<td>
<div align="center">meaning</div>
</td>
<td bgcolor="#E7EFCE"> </td>
<td>
<div align="center">rank</div>
</td>
<td>
<div align="center">symbol</div>
</td>
<td>
<div align="center">meaning</div>
</td>
</tr>
<tr>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">( )</div>
</td>
<td>
<div align="center">grouping</div>
</td>
<td> </td>
<td>
<div align="center">5</div>
</td>
<td>
<div align="center">+ -</div>
</td>
<td>
<div align="center">add, subtract</div>
</td>
</tr>
<tr>
<td>
<div align="center">2</div>
</td>
<td>
<div align="center">(-) (+) not</div>
</td>
<td>
<div align="center">sign, negation</div>
</td>
<td> </td>
<td>
<div align="center">6</div>
</td>
<td>
<div align="center">> < <= >=</div>
</td>
<td>
<div align="center">compare</div>
</td>
</tr>
<tr>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">^</div>
</td>
<td>
<div align="center">exponent (and root)</div>
</td>
<td> </td>
<td>
<div align="center">7</div>
</td>
<td>
<div align="center">= <></div>
</td>
<td>
<div align="center">equal, not equal</div>
</td>
</tr>
<tr>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">* / mod</div>
</td>
<td>
<div align="center">multiply, divide</div>
</td>
<td> </td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">and, or</div>
</td>
<td>
<div align="center">conjunction, alternation</div>
</td>
</tr>
</table>
</li>
<li>Operations of equal priority are evaluated left to right. Thus, 2 * 3 / 4 is the same as (2 * 3) / 4, but not 2 * (3/4).</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>aa</h5>
<p>bb.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>cc.</p>
</li>
<li>
<p>cc.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<tr>
<td bgcolor=#FFFFFF colspan="4">
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="#L5"></a><a href="#top">New heading.</a></h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<li> <span onclick="OLine(Sec13)"> <a href="#">Point 1.</a></span></li>
<div class="off" ID="Sec13">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top"> </td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">text.</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>text.</p>
</li>
<li>
<p> Text</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">Text</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Text</p>
</li>
<li>
<p> Text</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<div class="off" ID="Sec14"></div>
<tr>
<td bgcolor=#FFFFFF colspan="4">
<tr>
<td bgcolor=#DFE2ED colspan="4">
<div align="center"> <font size="2"><br>
</font><font size=+1><a href="file:///PowerBook%20HD/JS%20%C4/%20"><font size="2">Back</font></a><font size="2"> | <a href="file:///PowerBook%20HD/JS%20%C4/%20">Index</a> | <a href="file:///PowerBook%20HD/JS%20%C4/%20">Next</a></font></font></div>
<div align="center">
<hr size="1" noshade>
<p><font size="2">Copyright © 2001 <a href="mailto:factivity at hotmail.com">Roger Kenyon</a>. All rights reserved.<br>
http://www.google.com/ </font></p>
</div>
</td>
</table>
</td>
</body>
</html>
-------------- next part --------------
<html>
<head>
<title>Operations</title>
<STYLE><!--
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word[1]="<b>Yes, -40 and 163.</b><hr size='1' noshade><ol><li>Use the strategy of attempt and adjust. <br></li><li>Substitute the temperatures given for the variables in appropriate formula. <br><br>C = (F - 32) * 5 / 9 <br> C = (325 - 32) * 5 / 9 <br>C = (293) * 5 / 9 <br>C = 1465 / 9 <br> C = 162.777 . . .</li>";
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</head>
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<P>
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<div align="center"><a name="top"></a><a href="../index.html">Home</a> • <a href="subjects.htm">Subjects</a>
<h2>Operations</h2>
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<div align="left">Fahrenheit = Celsius * 9 / 5 + 32 and Celsius = (Fahrenheit - 32) * 5 / 9. At what temperature are the scales the same and what is the Celsius equivalent of a 325° F oven?</div>
<ol>
<li><a href="#" onClick="popNote(0);return false">-44, 178</a></li>
<li><a href="#" onClick="popNote(0);return false">-40, 177</a></li>
<li><a href="#" onClick="popNote(0);return false">-44, 177</a></li>
<li><a href="#" onClick="popNote(1);return false">-40, 163</a></li>
</ol>
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<div align="center">degrees Celsius</div>
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<div align="center">degrees Fahrenheit</div>
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<input name="celsius" onKeyUp="convert('C')">
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<input name="fahrenheit" onKeyUp="convert('F')">
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<p><br>
Upon completion of this strand, the learner should be able to apply to apply arithmetic operations and check accuracy of their outcome, such as using estimation to determine the reasonableness of a multiplication. </p>
</div>
<ol>
<li>
<div align="left"><a href="#L1">Logical and relational operators determine the truth value of an expression</a></div>
</li>
<li>
<div align="left"><a href="#L2">Linear numerical operators produce a straight line when graphed</a></div>
</li>
<li>
<div align="left"><a href="#L3">Exponential numerical operators produce a curve when graphed</a></div>
</li>
<li>
<div align="left"><a href="#L4">Arithmetic operations have properties and precedence</a></div>
</li>
</ol>
<div align="left">
<p>Key concepts: condition, conjunction, disjuction, denial, equality, inequality, addition, subtraction, multiplication, division, integer division, modulo, inverse operation, estimation, base, exponent, square, cube, exponential notation, scientific notation, root, involution, evolution, logarithm, common base, natural base, association, commutation, distribution, order of operations, grouping indicators.</p>
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<h4><a name="L1"></a><a href="#top">Logical and relational operators determine the truth value of an expression.</a> </h4>
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<li> <span onclick="OLine(Sec1)"> <a href="#">Logical operators affect whether an expression is true or false.</a></span></li>
<div class="off" ID="Sec1">
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<h5>condition</h5>
<p>Condition [IF]: the consequent is true on condition that the antecedent is true.</p>
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<ol>
<li>
<p>If [antecedent] the electricity is off, then [consequent] the lamp is not on.</p>
</li>
<li>Statements of fact can be stated as conditional expressions.
<p> <b>Bob likes gold</b> becomes <b>if something is gold, then Bob likes it</b>, but not <b>if Bob likes it, then it is gold</b>. He also likes pizza, but that doesn't make pizza into gold.</p>
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<h5>conjunction</h5>
<p>Conjunction [AND]: if all the items joined together (conjuncts) are true, then the proposition as a whole is true. If any conjunct is false, then the proposition as a whole is false.</p>
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<li>
<p>Statement A is true by itself. Statement B is true, also on its own. Therefore, the statement <b>A and B</b> is true.</p>
</li>
<li>
<p>Today is Tuesday. The sun is Tuesday. Therefore, today is Wednesday and the sun is shining.</p>
</li>
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<h5>disjunction</h5>
<p>Disjunction [OR]: if any alternative (disjunct) is true, then the proposition as a whole is true. When all disjuncts are false, then the proposition as a whole is false.</p>
</div>
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<ol>
<li>
<p>Statement A is true by itself. Statement B is false. Therefore, the statement <b>A or B</b> is true.</p>
</li>
<li>
<p>You are reading. You are asleep. Therefore, you are reading or you are asleep.</p>
</li>
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<h5>denial</h5>
<p>Denial [NOT]: If an item is true, then its denial is false; if an item is false, then its denial is true.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>Treasure is in one of three caves. Above each cave is an inscription. Only one inscription is true, but <a href="#" onClick="popNote(2);return false">which</a>?
<table width="95%" border="0" cellspacing="2" cellpadding="5">
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<div align="center">cave A</div>
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<div align="center">cave B</div>
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<div align="center">cave C</div>
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<div align="center"><b>treasure here</b></div>
</td>
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<div align="center"><b>treasure not here</b></div>
</td>
<td height="22">
<div align="center"><b>treasure not in cave A</b></div>
</td>
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<br>
</li>
<li>Double denial has the same value as an affirmation (two negatives equal a positive).
<p>on = on</p>
<p>not on = off</p>
<p>not not on = on</p>
</li>
</ol>
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<br>
<li> <span onclick="OLine(Sec2)"> <a href="#">Relational operators compare one term with another.</a> </span> </li>
<div class="off" ID="Sec2">
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<h5>equality</h5>
<p>The equals sign (that is, =) indicates that the expressions have the same value.</p>
</div>
</td>
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<ol>
<li>
<p>Equality is like a tetter totter or balance scale.</p>
</li>
<li>
<p>The comparison 2+2 = 3 is false, but correctly constructed.</p>
</li>
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<h5>inequality</h5>
<p>Inequality signs indicate that the expressions do not have the same value.</p>
</div>
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<td width="75%" height="75">
<ol>
<li>The angle opens to the larger value.
<table border="0" cellspacing="2" cellpadding="5">
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<td>greater than</td>
<td>3 > 2</td>
<td> </td>
<td>less than </td>
<td>2 < 3</td>
</tr>
<tr>
<td>not equal to</td>
<td>1 <> 2</td>
<td> </td>
<td> </td>
<td> </td>
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</table>
</li>
<li>Equality and inequality may be used disjunctively.
<p>Greater than or equal to: 27 >= 3</p>
less than or equal to: 2 <= 1+1</li>
</ol>
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<h4><a name="L2"></a><a href="#top">Linear numerical operators produce a straight line when graphed.</a></h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<li> <span onclick="OLine(Sec3)"> <a href="#"> Linear operations add, subtract, multiply, and divide.</a></span></li>
<div class="off" ID="Sec3">
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<h5>addition</h5>
<p>Addition is the operation of combining two sets.</p>
<p>Example showing Y = X + 2</p>
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<div align="center">9</div>
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<div align="center">8</div>
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<div align="center">7</div>
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<div align="center">6</div>
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<div align="center">5</div>
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<div align="center">4</div>
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<div align="center">3</div>
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<td bgcolor="#333333"> </td>
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<div align="center">2</div>
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<div align="center">1</div>
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<div align="center">0</div>
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<div align="center">1</div>
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<div align="center">2</div>
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<div align="center">3</div>
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<div align="center">5</div>
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<div align="center">6</div>
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<div align="center">7</div>
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<div align="center">8</div>
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<div align="center">9</div>
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<p> </p>
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<ol>
<li>
<p> Addend plus addend equals sum. A + B = C. 3 + 2 = 5</p>
</li>
<li>
<p>Only addends with the same denomination can be added; such as $14 + $0.96, but not $14 + 88 apples.</p>
</li>
<li>
<p>The pencil-and paper procedure for division is align, combine, and regroup.</p>
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<div align="center">align</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">combine</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">regroup</div>
</td>
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<div align="center">Put one addend under the other addend, lining up like place values.</div>
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<td width="5%"> </td>
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<div align="center">For each place value, starting with the lowest, combine the digits from the different addends.</div>
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<td width="5%"> </td>
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<div align="center">Regroup to the left as needed.</div>
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<div align="center">7</div>
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<div align="center">5</div>
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<div align="center">3</div>
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<div align="center">+</div>
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<div align="center">9</div>
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<div align="center">2</div>
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<div align="center">5</div>
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<div align="center">+</div>
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<div align="center">9</div>
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<div align="center">2</div>
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<td> </td>
<td colspan="3">
<div align="center">----------</div>
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<td> </td>
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<div align="center">5</div>
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<div align="center">1</div>
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<div align="center">7</div>
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<div align="center">5</div>
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<div align="center">3</div>
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<div align="center">+</div>
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<div align="center">9</div>
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<div align="center">2</div>
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<td> </td>
<td colspan="3">
<div align="center">----------</div>
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<div align="center">8</div>
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<div align="center">4</div>
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<div align="center">5</div>
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<h5>subtraction</h5>
<p>Subtraction is the operation of matching two sets and canceling members.</p>
</div>
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<ol>
<li>
<p>Minuend minus subtrahend equals difference. A - B = C; 3 - 2 = 1. If each of 30 students removes 2 chairs from the store-room holding 66 chairs, match two chairs per student and cancel: 66 chairs - (30 students * 2 chairs per student).</p>
</li>
<li>
<p>Minuends and subtrahends must have the same denomination; such as 7 m - 30 cm, but not 11 apples - 4 oranges.</p>
</li>
<li>
<p>The pencil-and paper procedure for division is align, regroup, and remove.</p>
<table cellspacing="2" cellpadding="5" width="95%">
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<div align="center">align</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">regroup</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">remove</div>
</td>
</tr>
<tr valign="top">
<td width="30%" bgcolor="#FFCE9C">
<div align="center">Place like units under like units.</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">If the minuend is smaller than the subtrahend, regroup from the minuend in the next higher place value.</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">Starting with the lowest place value, find the difference between each place value pair.</div>
</td>
</tr>
<tr valign="top">
<td height="215" width="30%" bgcolor="#FFCE9C">
<div align="center">
<table width="95%" border="0" cellspacing="0" cellpadding="5">
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td>
<div align="center">7</div>
</td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">2</div>
</td>
</tr>
<tr>
<td>
<div align="center">-</div>
</td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">8</div>
</td>
</tr>
</table>
</div>
</td>
<td height="215" width="5%"> </td>
<td height="215" width="30%" bgcolor="#FFCE9C">
<div align="center">
<table width="95%" border="0" cellspacing="0" cellpadding="5">
<tr>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">1</div>
</td>
</tr>
<tr>
<td> </td>
<td>
<div align="center">7</div>
</td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">2</div>
</td>
</tr>
<tr>
<td>
<div align="center">-</div>
</td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">8</div>
</td>
</tr>
<tr>
<td> </td>
<td colspan="3">
<div align="center">----------</div>
</td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">4</div>
</td>
</tr>
</table>
</div>
</td>
<td height="215" width="5%"> </td>
<td height="215" width="30%" bgcolor="#FFCE9C">
<div align="center">
<table width="95%" border="0" cellspacing="0" cellpadding="5">
<tr>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">1</div>
</td>
</tr>
<tr>
<td> </td>
<td>
<div align="center">7</div>
</td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">2</div>
</td>
</tr>
<tr>
<td>
<div align="center">-</div>
</td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">8</div>
</td>
</tr>
<tr>
<td> </td>
<td colspan="3">
<div align="center">----------</div>
</td>
</tr>
<tr>
<td> </td>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">0</div>
</td>
<td>
<div align="center">4</div>
</td>
</tr>
</table>
</div>
</td>
</tr>
</table>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>multiplication</h5>
<p>Multiplication is the operation of matching a set for each member of another set.</p>
<p>What is the product of (A - X) * (B - X) * (C - X) * . . . * (Z - X), where the letters stand for any number or other? Answer: zero. In the pattern is X - X, which is 0 and 0 times all the rest is 0.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Factor times factor equals product. A * B = C; 3 * 2 = 6. If each of 30 students is given a pack of 12 crayons, then the set of 12 is matched with each member of the set of 30.</p>
</li>
<li>
<p>If zero is a factor, the product is zero. The product of 1 and any factor is always the other factor; that is, 1 * A = A. </p>
</li>
<li>
<p>Denomination of the product depends how the factors are denominated, such as 7 cm * 3 = 21 cm, 7 cm * 3 cm = 21 cm^2 (meaning 21 square centimeters).</p>
</li>
<li>
<p>The pencil-and paper procedure for division is align, multiply, and combine.</p>
<table cellspacing="2" cellpadding="5" width="95%">
<tr>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">align</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">multiply</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">combine</div>
</td>
</tr>
<tr valign="top">
<td width="30%" bgcolor="#FFCE9C">
<div align="center">Align one factor under the other factor, lining up digits by place value.</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">Find the product of each digit of the first factor times each digit of the second factor; regroup as necessary.</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">Add up the partial products.</div>
</td>
</tr>
<tr valign="top">
<td height="215" width="30%" bgcolor="#FFCE9C">
<div align="center">
<table width="95%" border="0" cellspacing="0" cellpadding="5">
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">7</div>
</td>
</tr>
<tr>
<td>
<div align="center">*</div>
</td>
<td> </td>
<td>
<div align="center">2</div>
</td>
<td>
<div align="center">1</div>
</td>
</tr>
<tr>
<td> </td>
<td colspan="3"> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td colspan="3"> </td>
</tr>
</table>
</div>
</td>
<td height="215" width="5%"> </td>
<td height="215" width="30%" bgcolor="#FFCE9C">
<div align="center">
<table width="95%" border="0" cellspacing="0" cellpadding="5">
<tr>
<td> </td>
<td> </td>
<td>
<div align="center">1</div>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">7</div>
</td>
</tr>
<tr>
<td>
<div align="center">*</div>
</td>
<td> </td>
<td>
<div align="center">2</div>
</td>
<td>
<div align="center">1</div>
</td>
</tr>
<tr>
<td> </td>
<td colspan="3">
<div align="center">----------</div>
</td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">7</div>
</td>
</tr>
<tr>
<td> </td>
<td>
<div align="center">9</div>
</td>
<td>
<div align="center">4</div>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td colspan="3"> </td>
</tr>
<tr>
<td height="20"> </td>
<td height="20"> </td>
<td height="20"> </td>
<td height="20"> </td>
</tr>
</table>
</div>
</td>
<td height="215" width="5%"> </td>
<td height="215" width="30%" bgcolor="#FFCE9C">
<div align="center">
<table width="95%" border="0" cellspacing="0" cellpadding="5">
<tr>
<td> </td>
<td> </td>
<td>
<div align="center">1</div>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">7</div>
</td>
</tr>
<tr>
<td>
<div align="center">*</div>
</td>
<td> </td>
<td>
<div align="center">2</div>
</td>
<td>
<div align="center">1</div>
</td>
</tr>
<tr>
<td> </td>
<td colspan="3">
<div align="center">----------</div>
</td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">7</div>
</td>
</tr>
<tr>
<td> </td>
<td>
<div align="center">9</div>
</td>
<td>
<div align="center">4</div>
</td>
<td> </td>
</tr>
<tr>
<td>
<div align="center">+</div>
</td>
<td colspan="3">
<div align="center">----------</div>
</td>
</tr>
<tr>
<td height="20"> </td>
<td height="20">
<div align="center">9</div>
</td>
<td height="20">
<div align="center">8</div>
</td>
<td height="20">
<div align="center">7</div>
</td>
</tr>
</table>
</div>
</td>
</tr>
</table>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>division</h5>
<p> Division is the operation of separating a quantity into groups evenly.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Dividend divided by divisor equals quotient. A / B = C; 485 / 32 = 15.15625 or 15 5/32.</p>
</li>
<li>
<p>The divisor indicates how many groups into which dividend quantity is to be put. Division by zero is not possible since a quantity (the dividend) cannot both exist and yet be put into no groups (the 0 divisor).</p>
</li>
<li>
<p>When the dividend is denominated and the divisor is not, the quotient is denominated like the dividend. 21 kg / 3 = 7 kg. When both dividend and divisor are denominated, the quotient is not denominated. 21 kg / 7 kg = 3.</p>
</li>
<li>
<p>The pencil-and paper procedure for division is multiply, subtract, and repeat.</p>
<table cellspacing="2" cellpadding="5" width="95%">
<tr>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">multiply</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">subtract</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">repeat</div>
</td>
</tr>
<tr valign="top">
<td width="30%" bgcolor="#FFCE9C">
<div align="center">Working from left to right with digits of the dividend, find the largest multiple of the divisor whose product is not greater than the dividend digits.</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">Subtract and bring down the next dividend digit or digits.</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">Continue until the difference is 0 or a pattern in the quotient repeats.</div>
</td>
</tr>
<tr valign="top">
<td height="215" width="30%" bgcolor="#FFCE9C">
<div align="center">
<table width="95%" border="0" cellspacing="0" cellpadding="5">
<tr>
<td width="17%"> </td>
<td> </td>
<td width="17%">
<div align="center">0</div>
</td>
<td width="17%">
<div align="center">2</div>
</td>
<td width="18%"> </td>
<td width="21%"> </td>
</tr>
<tr>
<td height="4"></td>
<td bgcolor="#9C9C9C" height="4" width="4" colspan="5"></td>
</tr>
<tr>
<td width="17%">
<div align="center">5</div>
</td>
<td bgcolor="#9C9C9C"> </td>
<td width="17%">
<div align="center">1</div>
</td>
<td width="17%">
<div align="center">3</div>
</td>
<td width="18%">
<div align="center">9.</div>
</td>
<td width="21%"> </td>
</tr>
<tr>
<td width="17%"> </td>
<td> </td>
<td width="17%">
<div align="center">1</div>
</td>
<td width="17%">
<div align="center">0</div>
</td>
<td width="18%"> </td>
<td width="21%"> </td>
</tr>
</table>
</div>
</td>
<td height="215" width="5%"> </td>
<td height="215" width="30%" bgcolor="#FFCE9C">
<div align="center">
<table width="95%" border="0" cellspacing="0" cellpadding="5">
<tr>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">2</div>
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td height="4"></td>
<td colspan="5" bgcolor="#9C9C9C" height="4"></td>
</tr>
<tr>
<td>
<div align="center">5</div>
</td>
<td bgcolor="#9C9C9C"> </td>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">9.</div>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">0</div>
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td colspan="2">
<div align="center">------</div>
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">9</div>
</td>
<td> </td>
</tr>
</table>
</div>
</td>
<td height="215" width="5%"> </td>
<td height="215" width="30%" bgcolor="#FFCE9C">
<div align="center">
<table width="95%" border="0" cellspacing="0" cellpadding="5">
<tr>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">2</div>
</td>
<td>
<div align="center">7.</div>
</td>
<td>
<div align="center">8</div>
</td>
</tr>
<tr>
<td height="4"></td>
<td colspan="5" bgcolor="#9C9C9C" height="4"></td>
</tr>
<tr>
<td>
<div align="center">5</div>
</td>
<td bgcolor="#9C9C9C"> </td>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">9.</div>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">0</div>
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td colspan="2">
<div align="center">------</div>
</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">9</div>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">5</div>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td colspan="2">
<div align="center">------</div>
</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">0</div>
</td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">0</div>
</td>
</tr>
<tr>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td colspan="2">
<div align="center">------</div>
</td>
</tr>
<tr>
<td height="28"> </td>
<td height="28"> </td>
<td height="28"> </td>
<td height="28"> </td>
<td height="28"> </td>
<td height="28">
<div align="center">0</div>
</td>
</tr>
</table>
</div>
</td>
</tr>
</table>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>integer division</h5>
<p> Integer division (“div”) is dividing to return only the integer portion; also called “divide and truncate”.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>The dividend is divided by the divisor, ignoring any fractional portion, resulting in just the whole part.</p>
</li>
<li>
<p>485 div 32 = 15.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>modulo</h5>
<p>Modulo (“mod”) is dividing to find only an integer remainder (called the “modulus”). The dividend is modulated (divided) by the divisor, ignoring the whole part, resulting in just the remainder expressed as an integer.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Dividend modulo divisor equals modulus. A mod B = C; 485 mod 32 = 5. </p>
</li>
<li>
<p>To find the modulus with a calculator, divide [37/8 = 4.625], ignore the decimal, multiply the integer in the quotient by the divisor [4*8 = 32], then subtract the product from the dividend [37-32 = 5]. </p>
</li>
<li>
<p>Modulo is useful in finding a place in a cycle, such as the day of the week 100 days from now; 100 mod 7 = 2 so it will be 2 days later in the week than today. </p>
</li>
<li>
<p>If A mod B = 0, then B is said to be an aliquot factor of A. That is, a divisor is said to be an aliquot factor of a dividend if the operation leaves no remainder. For example, 6 is an aliquot factor of both 18 and 42. </p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<div align="left"><br>
</div>
<li> <span onclick="OLine(Sec4)"> <a href="#"> The results of arithmetic operations can be checked for accuracy.</a></span></li>
<div class="off" ID="Sec4">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5> inverse operation</h5>
<p>If [A operation B] = C, then [B inverse operation C] = A.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Addition and subtraction are inverse operations. If 286 - 97 = 189, then 97 + 189 = 286.</p>
</li>
<li>
<p>Multiplication and division are inverse operations. If 425 / 25 = 17, then 25 * 17 = 425.</p>
</li>
<li>
<p>Power and root are inverse operations. If 3^2 = 9, then 9^(1/2) = 3.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5> estimation</h5>
<p>To estimate, round numbers for easier mental math operations.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>In 8 * 47, round 47 to 50. Since 8 * 50 = 400, the answer will be almost 400.</p>
</li>
<li>
<p>Rounding up to a higher the place value is easier to calculate mentally, but leads to a less accurate estimate. </p>
<ol>
<li>
<p>461 * 4 = 1844, precise answer</p>
</li>
<li>
<p>450 * 4 = 1800, rounded to tens place</p>
</li>
<li>
<p>500 * 4 = 2000, rouded to hundreds place</p>
</li>
</ol>
</li>
</ol>
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<p align="center">
</td>
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<p align="center">
</td>
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</table>
</div>
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<td colspan="4" align="center" valign="middle" bordercolor="#FFFFFF" bgcolor="#FFFFFF">
<tr bgcolor="#FFFFFF" bordercolor="#FFFFFF">
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="L3"></a><a href="#top">Exponential numerical operators produce a curve when graphed</a></h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<li> <span onClick="OLine(Sec5)"> <a href="#">Exponentiation is the process of repeatedly multiplying a number by itself.</a></span></li>
<div class="off" id="Sec5">
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</td>
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<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>base, exponent</h5>
<p>Repeatedly adding equal quantities can be expressed by multiplication (e.g.: 7 + 7 + 7 = 7 * 3). So too, repeatedly multiplying equal quantities can be expressed with an exponent (e.g.: 8 * 8 * 8 = 8^3).</p>
<p>Example showing Y = X^2</p>
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<div align="center">9</div>
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<div align="center">8</div>
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<div align="center">7</div>
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<td bgcolor="#FFCE9C" width="12"> </td>
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<div align="center">6</div>
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<td width="12"> </td>
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<td bgcolor="#FFCE9C" width="12"> </td>
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<div align="center">5</div>
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<td width="12"> </td>
<td width="12"> </td>
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<tr>
<td bgcolor="#FFCE9C" width="12"> </td>
<td bgcolor="#313131" width="12"> </td>
<td bgcolor="#FFCE9C" width="12"> </td>
<td bgcolor="#FFCE9C" width="12">
<div align="center">4</div>
</td>
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<td bgcolor="#313131" width="12"> </td>
<td width="12"> </td>
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<td bgcolor="#FFCE9C" width="12"> </td>
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<div align="center">3</div>
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<div align="center">2</div>
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<div align="center">1</div>
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<div align="center">-3</div>
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<div align="center">-2</div>
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<div align="center">-1</div>
</td>
<td width="12">
<div align="center">0</div>
</td>
<td width="12">
<div align="center">1</div>
</td>
<td width="12">
<div align="center">2</div>
</td>
<td width="12">
<div align="center">3</div>
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</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Base to the power of exponent equals product. A^B = C; 3^2 = 9.</p>
</li>
<li>
<p>The base is the number used as a factor and the exponent indicates the number of factors. In 8^3, the base is 8, the exponent is 3, and the caret symbol “^” indicates exponentiation. In this case 8 is said to be raised to the power of 3. </p>
</li>
<li>
<p>A number raised to the power of two (e.g.: 5^2) is said to be “square”. A number raised to the power of three (e.g.: 5^3) is said to be “cube”.</p>
</li>
<li>
<p>The caret, superscript, and scientific formats express interchangeable values. Example: radius of Earth is 5 Mm = 5 * 10^6 m = 5 * 106 m = 5.00e+6 m. Caret notation has the advantage of being directly usable on most computer spreadsheets.</p>
</li>
</ol>
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</td>
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<div align="center">
<h5>square, cube</h5>
<p>Repeatedly adding equal quantities can be expressed by multiplication (e.g.: 7 + 7 + 7 = 7 * 3). So too, repeatedly multiplying equal quantities can be expressed with an exponent (e.g.: 8 * 8 * 8 = 8^3).</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>A number with a decimal of less than half will round up 4 out of 9 times (that is, 0.1, 0.2, 0.3, and 0.4 round to 1). A number with a decimal of half or more will round up 5 out of 9 times (that is, 0.5, 0.6, .7 , 0.8, or 0.9 round to 2).</p>
</li>
<li>
<p>In the rule of rounding up and down: decimals of half [0.5] preceded by an even number are rounded down. If preceded by an odd number, they are rounded up.</p>
</li>
</ol>
</td>
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<td valign="top"> </td>
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<p>
</td>
<td width="75%"></td>
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</div>
<br>
<li> <span onClick="OLine(Sec6)"> <a href="#">There are various rules of exponent arithmetic.</a> </span> </li>
<div class="off" id="Sec6">
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<h5>exponential notation</h5>
<p> There are rules to simplify the process of carrying out an arithmetic operation between values expressed in exponential notation.</p>
<p>8 * 32 = 256</p>
<p>2^3 * 2^5 = 256</p>
</div>
</td>
<td width="75%" height="76">
<ol>
<li> To multiply two powers of the same base or number add the exponents.
<table width="95%" border="0" cellspacing="2" cellpadding="5">
<tr>
<td>Format</td>
<td>base^A * base^B = base^(A + B)</td>
</tr>
<tr>
<td>Example</td>
<td>2^3 * 2^5 = 2^(3+5) = 2^8 = 256</td>
</tr>
</table>
<br>
</li>
<li> To multiply two numbers raised to the same power, add the bases.
<table width="95%" cellpadding="5" cellspacing="2" border="0">
<tr>
<td>Format</td>
<td>A^exponent * B^exponent = (A * B)^exponent</td>
</tr>
<tr>
<td>Example</td>
<td>5^3 * 7^3 = (5*7)^3 = 42 875</td>
</tr>
</table>
<br>
</li>
<li>To subtract two powers of the same base or number, subtract the exponent of the divisor from the exponent of the dividend.
<table width="95%" border="0" cellspacing="2" cellpadding="5">
<tr>
<td>Format</td>
<td>base^A / base^B = base^(A+B)</td>
</tr>
<tr>
<td>Example</td>
<td> 2^5 / 2^3 = 2^(5-3) = 2^2 = 4</td>
</tr>
</table>
<br>
</li>
<li> To raise the power of a number to a power, multiply the exponents.
<table width="95%" cellpadding="5" cellspacing="2" border="0">
<tr>
<td>Format</td>
<td>base^A^B</td>
</tr>
<tr>
<td>Example</td>
<td>4^2^3 = 4^(2*3) = 4^6 = 4096</td>
</tr>
</table>
<br>
</li>
<li>Any base or number to the power of 0 equals 1.
<table width="95%" border="0" cellspacing="2" cellpadding="5">
<tr>
<td>Format</td>
<td>base^0 = 1</td>
</tr>
<tr>
<td>Example</td>
<td>57^0 = 1; also 0^0 = 1</td>
</tr>
</table>
<br>
</li>
<li> Any base or number to the power of 1 equals that same base or number.
<table width="95%" cellpadding="5" cellspacing="2" border="0">
<tr>
<td>Format</td>
<td>base^1 = base</td>
</tr>
<tr>
<td>Example</td>
<td>57^1 = 57; also 0^1 = 0</td>
</tr>
</table>
<br>
</li>
<li>To calculate the power of a common fraction, raise the numerator and denominator to the power.
<table width="95%" border="0" cellspacing="2" cellpadding="5">
<tr>
<td>Format</td>
<td>(numerator/denominator)^A = numerator^A / denominator^A</td>
</tr>
<tr>
<td>Example</td>
<td> (3/5)^2 = (3 * 3) / (5 * 5) = 9/25 = 0.36</td>
</tr>
<tr>
<td> </td>
<td>Raising a fraction to a positive power produces a lesser amount since it finds a fraction of a fraction. Example: 0.04^2 = 0.0016.</td>
</tr>
</table>
<br>
</li>
<li> To raise a base or number to a negative power is to find the reciprocal of that base or number to the power.
<table width="95%" border="0" cellspacing="2" cellpadding="5">
<tr>
<td>Format</td>
<td>base^(-A) = 1/base^A provided the base is not 0</td>
</tr>
<tr>
<td>Example</td>
<td> 2^(-3) = 1/2^3</td>
</tr>
<tr>
<td> </td>
<td>Raising a fraction to a negative power produces a greater value than the original decimal since it finds the reciprocal of a fraction of a fraction. Example: 0.4^(-2) = 6.25.</td>
</tr>
</table>
<br>
</li>
<li>The exponent of any power of 10 indicates the number of zeros after the 1 in the result.
<table width="95%" cellpadding="5" cellspacing="2" border="0">
<tr>
<td>Format</td>
<td>10^N = 1 & N number of zeros</td>
</tr>
<tr>
<td>Examples</td>
<td>10^6 = 1 000 000; 10^(-3) = 1/1000</td>
</tr>
</table>
<br>
</li>
<li>A negative power of 10 is a fraction.
<table width="95%" cellpadding="5" cellspacing="2" border="0">
<tr>
<td>Format</td>
<td>10^(-N) = 1 / (1 & N number of zeros)</td>
</tr>
<tr>
<td>Examples</td>
<td>10^(-3) = 1/1000 </td>
</tr>
<tr>
<td> </td>
<td>Decimals can be expressed in terms of negative powers of 10. Example: 0.03 = 3 * 10^(-2) or in scientific notation as 3.00e-2.</td>
</tr>
</table>
</li>
</ol>
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<p align="center">
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<div align="center">
<h5>scientific notation</h5>
<p>Scientific notation is a type of exponential notation where the place value is expressed in powers of ten.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>Write the correct number of significant digits with one non-zero digit to the left of the decimal point, then multiplying the number by the appropriate power of 10 (positive or negative). </p>
<ol>
<li>
<p>Core temperature of the sun 20 000 000 °C = 2.00e+07 °C.</p>
</li>
<li>
<p>Thickness of notebook paper 0.0001 m = 1.00e-4 m. </p>
</li>
</ol>
</li>
<li>
<p>Negative numbers are indicated with a negative exponents, such as the mass of an electron 9.11e-31 kg. </p>
</li>
</ol>
</td>
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<p align="center">
</td>
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<tr bgcolor="#FFFFFF">
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<p align="center">
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<td width="75%"></td>
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</div>
<br>
<li> <span onClick="OLine(Sec7)"> <a href="#">Root finds the factor components of a number.</a> </span> </li>
<div class="off" id="Sec7">
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<div align="center">
<h5>root</h5>
<p>The root of a number is one of the equal factors of that number.</p>
</div>
</td>
<td width="75%" height="76">
<ol>
<li>Root may be written as a reciprocal exponent. For instance, if 2^5 = 32, then 32^(1/5) = 2.</li>
<li>Root finds factor components of a number. For instance, 32^(1/5) = 2 shows that five factors of 2 (that is, 2 * 2 * 2 * 2 * 2) equal 32. </li>
</ol>
<p> </p>
</td>
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<p align="center">
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<div align="center">
<h5>involution, evolution</h5>
<p>In mathematics, “involution” is raising a quantity to a power and “evolution” is the finding of roots. Both processes can be seen as “exponentiation”.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>The Nth root of a base is a number that if taken as a factor N times produces the base. If the Nth root is 2, then the root is “square” and if the Nth root is 3 then the root is “cube”.</p>
</li>
<li>
<p> If a regular cube has a volume of 343 units, then each edge of the cube measures 7 units since 343^(1/3) = 7.</p>
</li>
</ol>
</td>
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<p align="center">
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<p align="center">
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<td width="75%"></td>
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</div>
<br>
<li> <span onclick="OLine(Sec8)"> <a href="#">The logarithm of a number is the exponent to which the base must be raised to equal that number.</a></span></li>
<div class="off" ID="Sec8">
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<div align="center">
<h5>logarithm</h5>
<p>The number must be a positive numeric expression and the base must be a positive number other than 1.</p>
</div>
</td>
<td width="75%" height="76">
<ol>
<li>
<p>The integer part of a logarithm is called the “characteristic” and the decimal part of a logarithm is called the “mantissa”. Example: in log 228 = 2.357934847 the characteristic is 2 and the mantissa is 0.357934847.</p>
</li>
<li>
<p>Log 64 means 10^? = 64. Log 64 = 1.806179974. </p>
</li>
</ol>
<p> </p>
</td>
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<p align="center">
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<div align="center">
<h5>common base, natural base</h5>
<p>Logarithms which use 10 as the base are called common or Briggs logarithms.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>Logarithms were discovered by John Napier (Scottish; 1550 to 1617). Napier used what is called the natural or "e" base. Like þ and 2^(1/2), e is irrational: nonterminating and non-repeating. For practical purposes, e = 2.71828183. </p>
</li>
<li>
<p> Henry Briggs (English; 1561 to 1630) liked what he saw in Napier’s work, but preferred 10 as a base. These are called the natural and common bases of logarithms, respectively</p>
</li>
</ol>
</td>
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<p align="center">
</td>
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<p align="center">
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<td width="75%"></td>
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</table>
</div>
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<td colspan="4" align="center" valign="middle" bordercolor="#FFFFFF" bgcolor="#FFFFFF">
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="L4"></a><a href="#top">Arithmetic operations have properties and precedence.</a> </h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<li> <span onClick="OLine(Sec9)"> <a href="#">Arithmetic properties change an expression's form without changing its value.</a></span></li>
<div class="off" id="Sec9">
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<div align="center">
<h5>association</h5>
<p>Changing the grouping of elements does not change the value of the expression.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Association is a property of addition, multiplication, conjunction, and disjunction</p>
</li>
<li>
<p>(1 + 2) + 3 = 1 + (2 + 3); <br>
(2 * 3) * 4 = 2 * (3 * 4).</p>
</li>
<li>
<p>A and (B and C) = (A and B) and C; <br>
A or (B or C) = (A or B) or C.</p>
</li>
</ol>
</td>
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<p>
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<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>commutation</h5>
<p>The order of the operation does not alter the value of the result.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li> Commutation is a property of addition, multiplication, conjunction, and disjunction.</li>
<li>2 + 3 = 3 + 2; <br>
4 * 5 = 5 * 4.</li>
<li>A and B = B and A; <br>
A or B = B or A.</li>
</ol>
</td>
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<p>
</td>
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<div align="center">
<h5>distribution</h5>
<p>Distribution is a property linking addition with multiplication, conjunction with disjunction.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li> A * (B + C) = A * B + A * C.</li>
<li>A and (B or C) = (A and B) or (A and C); <br>
A or (B and C) = (A or B) and (A or C).</li>
</ol>
</td>
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<p>
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<p>
</td>
<td width="75%"></td>
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</div>
<br>
<li> <span onClick="OLine(Sec10)"> <a href="#">Operations are applied in a conventional order or precedence.</a></span></li>
<div class="off" id="Sec10">
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<div align="center">
<h5>order of operations</h5>
<p>An operator is a symbol describing the operation to apply to one or more operand. An operand is the data element on which an operation is applied.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li> Hierarchy of frequently used operations:
<table width="95%" border="0" cellspacing="2" cellpadding="5">
<tr bgcolor="#FFCE9C">
<td>
<div align="center">rank</div>
</td>
<td>
<div align="center">symbol</div>
</td>
<td>
<div align="center">meaning</div>
</td>
<td bgcolor="#E7EFCE"> </td>
<td>
<div align="center">rank</div>
</td>
<td>
<div align="center">symbol</div>
</td>
<td>
<div align="center">meaning</div>
</td>
</tr>
<tr>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">( )</div>
</td>
<td>
<div align="center">grouping</div>
</td>
<td> </td>
<td>
<div align="center">5</div>
</td>
<td>
<div align="center">+ -</div>
</td>
<td>
<div align="center">add, subtract</div>
</td>
</tr>
<tr>
<td>
<div align="center">2</div>
</td>
<td>
<div align="center">(-) (+) not</div>
</td>
<td>
<div align="center">sign, negation</div>
</td>
<td> </td>
<td>
<div align="center">6</div>
</td>
<td>
<div align="center">> < <= >=</div>
</td>
<td>
<div align="center">compare</div>
</td>
</tr>
<tr>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">^</div>
</td>
<td>
<div align="center">exponent (and root)</div>
</td>
<td> </td>
<td>
<div align="center">7</div>
</td>
<td>
<div align="center">= <></div>
</td>
<td>
<div align="center">equal, not equal</div>
</td>
</tr>
<tr>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">* / mod</div>
</td>
<td>
<div align="center">multiply, divide</div>
</td>
<td> </td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">and, or</div>
</td>
<td>
<div align="center">conjunction, alternation</div>
</td>
</tr>
</table>
</li>
<li>Operations of equal priority are evaluated left to right. Thus, 2 * 3 / 4 is the same as (2 * 3) / 4, but not 2 * (3/4).</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>grouping</h5>
<p>Grouping indicators can be used to change the order of operations.</p>
</div>
</td>
<td bgcolor="#E3EECC">
<ol>
<li>
<p>Groups are indicated by braces, brackets, or parentheses.</p>
<table cellspacing="2" cellpadding="5" width="95%">
<tr>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">brace</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">bracket</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">parenthesis</div>
</td>
</tr>
<tr valign="top">
<td width="30%" bgcolor="#FFCE9C">
<div align="center">{ }</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">[ ]</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center">( )</div>
</td>
</tr>
<tr valign="top">
<td width="30%" bgcolor="#FFCE9C">
<div align="center"> Set S = {6, 7, 8}</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center"> 9 * [87 - (6 + 5)] = 684</div>
</td>
<td width="5%"> </td>
<td width="30%" bgcolor="#FFCE9C">
<div align="center"> 27 / 3 + 6 <> 27 / (3 + 6)</div>
</td>
</tr>
</table>
</li>
<li>
<p>Expressions in the innermost pair of grouping indicators are evaluated first.</p>
</li>
</ol>
<p> </p>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<tr>
<td bgcolor=#FFFFFF colspan="4">
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="#L5"></a><a href="#top">New heading.</a></h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<li> <span onclick="OLine(Sec13)"> <a href="#">Point 1.</a></span></li>
<div class="off" ID="Sec13">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top"> </td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">text.</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>text.</p>
</li>
<li>
<p> Text</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">Text</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Text</p>
</li>
<li>
<p> Text</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<div class="off" ID="Sec14"></div>
<tr>
<td bgcolor=#FFFFFF colspan="4">
<tr>
<td bgcolor=#DFE2ED colspan="4">
<div align="center"> <font size="2"><br>
</font><font size=+1><a href="file:///PowerBook%20HD/JS%20%C4/%20"><font size="2">Back</font></a><font size="2"> | <a href="file:///PowerBook%20HD/JS%20%C4/%20">Index</a> | <a href="file:///PowerBook%20HD/JS%20%C4/%20">Next</a></font></font></div>
<div align="center">
<hr size="1" noshade>
<p><font size="2">Copyright © 2001 <a href="mailto:factivity at hotmail.com">Roger Kenyon</a>. All rights reserved.<br>
http://www.google.com/ </font></p>
</div>
</td>
</table>
</td>
</body>
</html>
-------------- next part --------------
<html>
<head>
<title>Integers</title>
<STYLE><!--
SPAN {font-size: 12pt; cursor: hand; color:blue; font-weight:bold}
.on {display: on; margin-left: 15; margin-top: 10; margin-bottom: 10}
A {text-decoration: none}
A:hover {color: red; text-decoration: underline}
H5 {font-size:12pt; text-transform:capitalize; margin-top: 5; margin-bottom:0; color:darkblue}
P {font-size:12pt}
--></STYLE>
<script language="JavaScript">
var popUpBttn="<input type='button' value='Close Window' onClick='window.close();'>"
word=new Array();
word[0]="<b>Consider the facts and try again.</b><hr size='1' noshade><ol><li>From the cardinality of one set and correspondence of sets, find the cardinality of the other set. </li><li>From amount A and correspondence B : C, find the second amount. </li><li>Second amount = A / B * C. </li>";
word[1]="<b>Yes, 40 and 90.</b><hr size='1' noshade><ol><li>Generalize the problem. <br>From amount A and correspondence B : C, find the second amount. </li><li>State the solution as an algorithm. <br>Second amount = A / B * C. </li><li>Translate the algorithm into javascript. <br>Function card(A,B,C) {outputField.value = A/B*C}</ol></li>";
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}
function card(A,B,C) {
outputField.value = A/B*C
}
//-->
</script>
</head>
<body bgcolor=#FFFFFF>
<P>
<tr>
<td>
<table width=100% border=0 cellpadding=5 cellspacing="0">
<tr bgcolor="#FFFFFF">
<td colspan="4">
<div align="center"><a name="top"></a><a href="../index.html">Home</a> • <a href="subjects.htm">Subjects</a>
<h2>Integers</h2>
</div>
<tr>
<td bgcolor=#819888 colspan="4">
<table width="100%" border="0" cellspacing="0" cellpadding="20" bgcolor="#E3EECC">
<tr>
<td colspan="1" bgcolor="#E7EFCE">
<div align="left">Suppose a classroom orders 60 books. How many bonus
stickers would the room receive if (a) the bonus is 3 stickers for
every second book ordered; (b) the bonus is 2 stickers for every
third book ordered?</div>
<ol>
<li><a href="#" onClick="popNote(1);return false">40,
90</a></li>
<li><a href="#" onClick="popNote(0);return false">90,
40</a></li>
<li><a href="#" onClick="popNote(0);return false">120,
180</a></li>
<li><a href="#" onClick="popNote(0);return false">180,
120</a></li>
</ol>
<table border="0" cellspacing="0" cellpadding="4" width="100%">
<tr>
<td>
<div align="center">amount</div>
</td>
<td>
<div align="center">correspondence</div>
</td>
<td bgcolor="#FFCE9C">
<div align="center">amount</div>
</td>
</tr>
<tr>
<td>
<div align="center">
<input name=in1 size="8" onKeyUp="card(parseInt(in1.value),parseInt(in2.value),parseInt(in3.value))">
</div>
</td>
<td>
<div align="center">
<input name=in2 size="4" onKeyUp="card(parseInt(in1.value),parseInt(in2.value),parseInt(in3.value))">
:
<input name=in3 size="4" onKeyUp="card(parseInt(in1.value),parseInt(in2.value),parseInt(in3.value))">
</div>
</td>
<td bgcolor="#FFCE9C" bordercolor="#FFCE9C">
<div align="center">
<input type=text name=outputField onFocus="this.blur()" size="8">
</div>
</td>
</tr>
</table>
</td>
</tr>
</table>
<tr bgcolor="#FFFFFF">
<td align="center" colspan="4">
<div align="left">
<p><br>
Upon completion of this strand, the learner should be able to apply whole
number concepts, such as factoring a number into its primes and finding
lowest common multiple. </p>
</div>
<ol>
<li>
<div align="left"><a href="#L1">Number indicates how many items are in
a set</a> </div>
</li>
<li>
<div align="left"><a href="#L2">An integer is a whole number higher or
lower than zero</a></div>
</li>
<li>
<div align="left"><a href="#L3">The value of a digit depends on its place
and base</a></div>
</li>
<li>
<div align="left"><a href="#L4">Integers can have various properties</a></div>
</li>
</ol>
<div align="left">
<p>Key concepts: set, correspondence, ordinal and cardinal numbers, numerals,
rounding, approximation, negative, positive, absolute value, digit, period
grouping, decimal system (base 10), binary system (base 2), abundant,
perfect, deficient, amicable, prime, composite, greatest common factor,
lowest common multiple. </p>
</div>
<!--major topic start -->
<tr>
<td bgcolor=#FFFFFF colspan="4">
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="#L1"></a><a href="#top">Number indicates how many items are
in a set.</a> </h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<!-- block begins -->
<li> <span onclick="OLine(Sec1)"> <a href="#">A set is a group of items that
are related somehow.</a></span></li>
<div class="off" ID="Sec1">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>set</h5>
<p>Members of a set are called elements and can be listed between braces, separated by commas.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Stereo speakers come two to a set.</p>
</li>
<li>
<p> Alphabet = {A, B, .C, .., Z}.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>correspond</h5>
<p>Members of one group can be matched up with elements of another group.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>{fingers} to {toes}.</p>
</li>
<li>
<p>If A = {1, 2, 3, 4} and B = {W, X, Y, Z}, then 1 corresponds to W, 2 corresponds to X, 3 corresponds to Y, and 4 corresponds to Z.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>correspondence ratio</h5>
<p>Sets can correspond 1 to 1, 1 to many, or any other ratio.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Four hub caps for every four tires is still a 1:1 correspondence. That is, 4:4 or any N:N is 1:1.</p>
</li>
<li>
<p>Multiplication uses one-to-many correspondence. Imagine 2 * 3 as AA * BBB. Each A corresponds to and will be replaced by BBB, so AA becomes BBB BBB.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<br>
<!-- end block -->
<!-- block begins -->
<li> <span onclick="OLine(Sec2)"> <a href="#">Amounts are written with numerals.</a>
</span> </li>
<div class="off" ID="Sec2">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="76" bgcolor="#FFCE9C">
<div align="center">
<h5>numeral</h5>
<p>Number is a concept of an amount; numeral is a written symbol
representing a number.</p>
</div>
</td>
<td width="75%" height="76">
<ol>
<li>
<p>A dozen is a number (amount).</p>
</li>
<li>
<p>“12” is a numeral (symbol).</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>cardinal</h5>
<p>A cardinal number is the number of how many items are in a set.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>Eight notes in an octave.</p>
</li>
<li>
<p>Twelve months in a year.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>ordinal</h5>
<p>Ordinal numbers specify order or position in a set.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>Third base in baseball.</p>
</li>
<li>
<p>Second half of the game.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<!-- end block -->
<tr bgcolor="#FFFFFF" bordercolor="#FFFFFF">
<td colspan="4" align="center" valign="middle" bordercolor="#FFFFFF" bgcolor="#FFFFFF">
<!--major topic stop -->
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="L2"></a><a href="#top">An integer is a whole number higher or
lower than zero.</a> </h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<!-- block begins -->
<li> <span onClick="OLine(Sec3)"> <a href="#">Integers may result from rounding
decimals.</a></span></li>
<div class="off" id="Sec3">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>rounding up</h5>
<p>In rounding upward, change a number to 10 if the digit to its
right is greater than or equal to 5.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p> 385.25 rounded to the nearest tenth is 385.3, to the nearest
unit is 385, to the nearest hundred is 400.</p>
</li>
<li>
<p> Rounding up when a decimal is 5 or more produces favors larger
numbers. As to statistical bias, </p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">Rounding up produces a statistical bias in favor
of large numbers. </div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>A number with a decimal of less than half will round up 4
out of 9 times while any number with a decimal of half or more
will round up 5 out of 9 times. Thus, 0.1, 0.2, 0.3, and 0.4
round to 1 while 0.5, 0.6, .7 , 0.8, or 0.9 round to 2.</p>
</li>
<li>
<p>In the rule of rounding up and down: decimals of half [0.5]
preceded by an even number are rounded down. If preceded by
an odd number, they are rounded up.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>approximation</h5>
<p>Rounded numbers are often used in approximations. An approximation
is a number that is close for a specific purpose, but not exact.
</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>The tank can hold about 50 L of fuel.</p>
</li>
<li>
<p>The perimeter of a circle is approximately 3 times greater
than the diameter of that circle.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<br>
<!-- end block -->
<!-- block begins -->
<li> <span onClick="OLine(Sec4)"> <a href="#">Integers can be lesser or greater
than zero.</a> </span> </li>
<div class="off" id="Sec4">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="76" bgcolor="#FFCE9C">
<div align="center">
<h5>negative</h5>
<p>Integers less than zero are negative.</p>
</div>
</td>
<td width="75%" height="76">
<ol>
<li>
<p>Temperature might range from a high of 12 degrees to a low
of -3 degrees. <br>
</p>
</li>
<li>Note: -3 and +3 are both three units away from zero.
<table width="75%" border="0" cellspacing="2" cellpadding="5">
<tr>
<td colspan="3">Negative</td>
<td> </td>
<td colspan="3">
<div align="right">positive</div>
</td>
</tr>
<tr>
<td>
<div align="center">-3</div>
</td>
<td>
<div align="center">-2</div>
</td>
<td>
<div align="center">-1</div>
</td>
<td>
<div align="center">0</div>
</td>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">2</div>
</td>
<td>
<div align="center">3</div>
</td>
</tr>
</table>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>positive</h5>
<p>Integers less than zero are positive.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>The positive sign may be omitted; thus 3 is +3.</p>
</li>
<li>
<p>If the lobby of a tall building is floor 0, then going up
is positive, while going down is negative.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
<p align="center">
<p align="center">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>absolute value</h5>
<p>Absolute value is the numerical offset from zero.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>-7 °C is 7 degrees offset from 0 °C so its absolute
value is 7.</p>
</li>
<li>Absolute value is conventionally written between vertical strokes,
|-7| = 7.</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<p>To add negative amounts, treat the negative sign as subtraction.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>If the numbers have the same sign, then find their sum and
use the same sign. (+7) + (+5) = 12 and (-7) + (-5) = -12.</p>
</li>
<li>
<p>If the numbers have different signs, find the difference and
use the sign of the number with the greatest absolute value.
(+7) + (-5) = 2 and (-7) + (+5) = -2.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center"> To subtract negative and positive numbers, change
the sign of the subtrahend, then proceed as in addition.</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p> (-5) - (-7) ¼ (- 5) + 7 = 2 and (-5) - (+7) ¼ (-5) - 7 =
-12.</p>
</li>
<li>
<p> 5 - (-7) ¼ 5 + 7 = 12 and 5 - (+7) ¼ 5 - 7 = -2.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<p>The product of two numbers with the same sign is positive; otherwise
it is negative.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>If numbers have the same sign, then their product is positive.
(2 * 3) and (-2 * -3) are both equal to 6.</p>
</li>
<li>
<p>If numbers have different signs, then their product is negative
(-2 * 3) and (2 * -3) are both equal to -6.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center"> The quotient of two numbers with the same sign
is positive; otherwise it is negative.</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>If the dividend and divisor have the same sign, then the quotient
is positive. Examples: (+7) / (+5) = 1.4 and (-7) / (-5) =
1.4.</p>
</li>
<li>
<p>If the dividend and divisor have different signs, the quotient
is negative. Examples: (+7) / (-5) = -1.4 and (-7) / (+5) =
-1.4.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<!-- end block -->
<!-- block begins -->
<p>
<!-- end block -->
</p>
<tr bgcolor="#FFFFFF" bordercolor="#FFFFFF">
<td colspan="4" align="center" valign="middle" bordercolor="#FFFFFF" bgcolor="#FFFFFF">
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="L3"></a><a href="#top">The value of a digit depends on its place
and base.</a> </h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<!-- block begins -->
<li> <span onClick="OLine(Sec5)"> <a href="#">The value of a digit depends
on its place in a numeral.</a></span></li>
<div class="off" id="Sec5">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>digit</h5>
<p>“Digit” means finger and comes from a time when people traditionally counted on their fingers.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li>
<p>Digits are arranged in place value. In any given base N, each place value is N times greater than the place value to its right. 2 = two units, 20 = two tens and no ones, and 200 = two hundreds, no tens, no ones in base ten.</p>
</li>
<li>The decimal point is to the right of the units place, even if the point is not written. Example: 27 = 27.0</li>
</ol>
</td>
</tr>
<tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>grouping period</h5>
<p>A period is a group of three orders: units, tens, and hundreds.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li> The names of periods are: units, thousands, millions, billions, trillions, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, and decillions.
<table width="95%" border="0" cellspacing="2" cellpadding="5">
<tr>
<td colspan="3">
<div align="center">millions</div>
</td>
<td colspan="3">
<div align="center">thousands</div>
</td>
<td colspan="3">
<div align="center">units</div>
</td>
<td> </td>
<td colspan="3">
<div align="center">thousandths</div>
</td>
</tr>
<tr>
<td height="22">
<div align="center">8</div>
</td>
<td height="22">
<div align="center">4</div>
</td>
<td height="22">
<div align="center">7</div>
</td>
<td height="22">
<div align="center">5</div>
</td>
<td height="22">
<div align="center">3</div>
</td>
<td height="22">
<div align="center">1</div>
</td>
<td height="22">
<div align="center">8</div>
</td>
<td height="22">
<div align="center">6</div>
</td>
<td height="22">
<div align="center">4</div>
</td>
<td height="22">
<div align="center">.</div>
</td>
<td height="22">
<div align="center">0</div>
</td>
<td height="22">
<div align="center">9</div>
</td>
<td height="22">
<div align="center">2</div>
</td>
</tr>
</table>
</li>
<li>Names of numbers follow a pattern: prefix + “-illion” to indicate period order after million. Bi- [“two” as in bicycle] + -illion = billion. Trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion.</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<br>
<!-- block begins -->
<li> <span onClick="OLine(Sec6)"> <a href="#">The value of a digit depends
on the base of the number system.</a></span></li>
<div class="off" id="Sec6">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>decimal number system</h5>
<p>Base 10 uses the digits 0 through 9 and each place value is
10 times greater than the place value to its right.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li> The value 1234.567 indicates
<table width="75%" border="0" cellspacing="2" cellpadding="5">
<tr>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">2</div>
</td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">4</div>
</td>
<td> </td>
<td>
<div align="center">5</div>
</td>
<td>
<div align="center">6</div>
</td>
<td>
<div align="center">7</div>
</td>
</tr>
<tr>
<td>
<div align="center">1000s</div>
</td>
<td>
<div align="center">100s</div>
</td>
<td>
<div align="center">10s</div>
</td>
<td>
<div align="center">1s</div>
</td>
<td>
<div align="center">.</div>
</td>
<td>
<div align="center">1/10s</div>
</td>
<td>
<div align="center">1/100s</div>
</td>
<td>
<div align="center">1/1000s</div>
</td>
</tr>
</table>
</li>
<li>Names of numbers follow a pattern: prefix + “-illion”
to indicate period order after million. BI [“two” as
in bicycle] + -illion = billion. Trillion, quadrillion, quintillion,
sextillion, septillion, octillion, nonillion, decillion.</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>binary number system</h5>
<p>Base 2 uses the digits 0 through 1 and each place value is 2
times greater than the place value to its right.</p>
</div>
</td>
<td width="50%" height="75" bgcolor="#E3EECC">
<ol>
<li> The value 1101 in base 2 indicates
<table width="75%" border="0" cellspacing="2" cellpadding="5">
<tr>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">0</div>
</td>
<td>
<div align="center">1</div>
</td>
<td> </td>
<td>
<div align="center">=</div>
</td>
<td>
<div align="center">13 in base 10</div>
</td>
</tr>
<tr>
<td>
<div align="center">8s</div>
</td>
<td>
<div align="center">4s</div>
</td>
<td>
<div align="center">2s</div>
</td>
<td>
<div align="center">1s</div>
</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
</table>
</li>
<li> Base 2 is often used in computer machine code since 1 and
0 correspond to the electricity on and off.</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<!-- end block -->
<!-- block begins -->
<!-- end block -->
<!--major topic start -->
<tr>
<td bgcolor=#FFFFFF colspan="4">
<tr>
<td bgcolor=#F7A563 colspan="4">
<h4><a name="L4"></a><a href="#top">Integers can have various properties.</a>
</h4>
<tr bgcolor="#E3EECC">
<td colspan="4" bgcolor="#E3EECC"> <br>
<!-- block begins -->
<li> <span onclick="OLine(Sec7)"> <a href="#">Positive integers can be abundant
or perfect.</a> </span> </li>
<div class="off" ID="Sec7">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr>
<td width="50%" valign="top" height="76" bgcolor="#FFCE9C">
<div align="center">
<h5>abundant</h5>
<p>An abundant number is a positive integer whose factors other
than itself add up to more than the integer.</p>
</div>
</td>
<td width="75%" height="76">
<ol>
<li>
<p> The factors of 24 are 1, 2, 3, 4, 6, 8, and 12, but the sum
of these factors is 36, which is in excess of 24 itself.</p>
</li>
<li>
<p>c.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>perfect</h5>
<p>A perfect number a is positive integer whose factors other than
itself add up to the integer.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>Since 28 can be evenly divided by 1, 2, 4, 7, 14 and 1 + 2
+ 4 + 7 + 14 = 28, it is a perfect number.</p>
</li>
<li>
<p>Perfect numbers get large quickly. After 28, the next couple
perfect numbers are 496 and 8128.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p>
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p>
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<br>
<!-- end block -->
<!-- block begins -->
<li> <span onclick="OLine(Sec8)"> <a href="#">Integers can be deficient or
amicable.</a> </span> </li>
<div class="off" ID="Sec8">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>deficient</h5>
<p>A deficient number is a positive integer whose factors other
than itself add up to less than the integer.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p> The factors of 16 are 1, 2, 4, and 8 which add up to 15 rather
than 16, so 16 is deficient.</p>
</li>
<li>
<p>c.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>amicable</h5>
<p>A pair of numbers is said to be amicable if their factors sum
to each other, such as 1184 and 1210.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p> The smallest amicable pair is 220 and 284. </p>
<ol>
<li>
<p>The sum of the factors of 220 are 1 + 2 + 4 + 5 + 10 +
11 + 20 + 22 + 44 + 55 + 110 = 284. </p>
</li>
<li>
<p>The sum of the factors of 284 are 1 + 2 + 4 + 71 + 142
= 220.</p>
</li>
</ol>
</li>
<li>
<p>More than 600 amicable pairs are known, many having over 30
digits.</p>
<ol>
<li>
<p> In 1634, French mathematician Pierre de Fermat discovered
the amicable pair: 17 296 and 18 416. </p>
</li>
<li>
<p>Around the same time, fellow countryman René Descartes
discovered the amicable pair: 9 363 584 and 9 437 056.</p>
</li>
</ol>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
</table>
</div>
<br>
<!-- end block -->
<!-- block begins -->
<li> <span onclick="OLine(Sec9)"> <a href="#">Integers can be prime or composite.</a>
</span> </li>
<div class="off" ID="Sec9">
<table width="100%" border="0" cellpadding="2" cellspacing="0" bordercolor="#FFFFFF" dwcopytype="CopyTableRow">
<tr>
<td valign="top" bgcolor="#FFFFFF"> </td>
<td width="75%" bgcolor="#FFFFFF"> </td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>prime</h5>
<p>A prime number is a quantity evenly divisible only by itself
and 1.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>Eratosthenes (Greek, 275 to 195 BC) found a way to sift out
prime numbers. To make the Sieve of Eratosthenes:</p>
<ol>
<li>
<p>Write the numbers 1 to any number N. Cross out 1; it is
not treated as a prime.</p>
</li>
<li>
<p>Circle the first remaining number, then cross out all
its multiples. Repeat this step until all numbers have
been circled or crossed out.</p>
</li>
<li>
<p>Thus, circle 2. It is a prime. Go through and cross out
all multiples of 2. Circle 3. It is a prime. Cross out
multiples of 3. Continue in this fashion.</p>
</li>
<li>
<p>The prime numbers under 50 are: 2, 3, 5, 7, 11, 13, 17,
19, 23, 29, 31, 37, 41, 43, 47.</p>
</li>
</ol>
</li>
<li>
<p>The number 2 is the only even prime. All numbers are either
even or odd. A number divisible by 2 with no fractional portion
is called even.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>composite</h5>
<p>Any composite number can be factored into its prime numbers
(prime factors).</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p> To factor a number, divide it by one prime number after the
other until it is reduced to a product of prime numbers. 30
= 2 * 15 = 2 * 3 * 5.</p>
</li>
<li>
<p>A composite number is "composed" of prime numbers,
such as 36 which is composed of 2 * 2 * 3 * 3. Prime factors
should be written smallest to largest.</p>
</li>
</ol>
</td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr bgcolor="#FFFFFF">
<td valign="top">
<p align="center">
</td>
<td width="75%"></td>
</tr>
<td valign="top">
<p align="center">
</td>
</tr>
<tr>
<td width="50%" valign="top" height="75" bgcolor="#FFCE9C">
<div align="center">
<h5>GCF</h5>
<p>The greatest common factor of two or more numbers is the highest
number that evenly divides all the numbers in that set.</p>
</div>
</td>
<td width="75%" height="75">
<ol>
<li>
<p>To find the GCF of two numbers, factor them and multiply the
factors they have in common (regardless of how many times these
factors appear in the factorization).</p>
<ol>
<li>
<p>24 = 2 * 2 * 2 * 3 </p>
</li>
<li>
<p>30 = 2 * 3 * 5</p>
</li>
<li>
<p>Both numbers have 2 and 3 as prime factors. </p>
</li>
<li>
<p>2 * 3 = 6, so GCF of {24, 30} = 6.</p>
</li>
</ol>
</li>
<li>
<p>GCF isn't just a common factor, but the highest factor in
common. 2 is a factor of both 24 and 30, but 6 is greatest
factor common to both 24 and 30.</p>
</li>
</ol>
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<h5>LCM</h5>
<p>The lowest common multiple of two or more numbers is the lowest
number that is evenly divisible by all numbers in that set.</p>
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</td>
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<ol>
<li>
<p>To find the LCM of two numbers, list their multiples until
a the first multiple appears in both list.</p>
<ol>
<li>
<p>The multiples of 4 are 4, 8, 12, 16, 20, 24, …</p>
</li>
<li>
<p>The multiples of 6 are 6, 12, 18, 24, ….</p>
</li>
<li>
<p>12 is the first multiple that appears. </p>
</li>
<li>
<p> LCM of {4, 6} = 12. </p>
</li>
</ol>
</li>
<li>
<p>LCM isn't just a common multiple, but the lowest multiple
in common. 24 is a multiple of both 4 and 6, but 12 is the
lowest that 4 and 6 have in common.</p>
</li>
</ol>
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<li> <span onclick="OLine(Sec10)"> <a href="#">Point 1.</a></span></li>
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<li> <span onclick="OLine(Sec11)"> <a href="#">Point 2.</a> </span> </li>
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<div align="center"> <font size="2"><br>
</font><font size=+1><a href="file:///PowerBook%20HD/JS%20%C4/%20"><font size="2">Back</font></a><font size="2">
| <a href="file:///PowerBook%20HD/JS%20%C4/%20">Index</a> | <a href="file:///PowerBook%20HD/JS%20%C4/%20">Next</a></font></font></div>
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<hr size="1" noshade>
<p><font size="2">Copyright © 2001 <a href="mailto:factivity at hotmail.com">Roger
Kenyon</a>. All rights reserved.<br>
http://www.google.com/ </font></p>
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