Associative, commutative, distributive properties

<h1>Text Preview</h1> <p>In the previous lesson we learned about some important number properties that are listed here. Let's take a short review</p> <p>Click the “Submit” button after entering your answer.</p> <p>Today, we have three important number properties to learn, so let’s continue with the lesson.</p> <p>Here, we have the interactive scale we used previously.</p> <p>On the left-hand pan are two identical weights. Each weight has the value <font size="+2" face="Times New Roman"><i>b</i> + <i>c</i></font>.</p> <p>To the side of the scale are several weights that have values of <font size="+2" face="Times New Roman"><i>b</i></font> or <font size="+2" face="Times New Roman"><i>c</i></font>.</p> <p>Move these weights to the right-hand pan until the scale is balanced.</p> <p>How many <font size="+2" face="Times New Roman"><i>b</i></font> weights are on the right-hand pan when the scale is balanced? Click the “Submit” button after entering your answer.</p> <p>There should be two <font size="+2" face="Times New Roman"><i>b</i></font> weights on the right-hand pan when the scale is balanced.</p> <p>How many <font size="+2" face="Times New Roman"><i>c</i></font> weights are on the right-hand pan when the scale is balanced? Click the “Submit” button after entering your answer.</p> <p>There also should be two <font size="+2" face="Times New Roman"><i>c</i></font> weights on the right-hand pan.</p> <p>When the scale is balanced, we see that there are two <font size="+2" face="Times New Roman"><i>b</i> + <i>c</i></font> weights on the left-hand side, and two <font size="+2" face="Times New Roman"><i>b</i></font> and two <font size="+2" face="Times New Roman"><i>c</i></font> weights on the right-hand side.</p> <p>Which of the following is an equation that represents what we see with the scale? Click the “Submit” button after selecting your answer.</p> <p>The equivalent equation is: <font size="+2" face="Times New Roman">2(<i>b</i> + <i>c</i>) = 2<i>b</i> + 2<i>c</i></font>. This equation is an example of the <b>distributive property</b>.</p> <p>We can show this property in a more general way by replacing the coefficient <font size="+2" face="Times New Roman">2</font> in our equation by any number <font size="+2" face="Times New Roman"><i>a</i>.</p> <p>Let’s try an example. Here we have an expression containing a term in parentheses.</p> <p>Use the distributive property to rewrite the expression without parentheses. Click the “Submit” button after entering your answer.</p> <p>Our example showed the distributive property used when the term in the parentheses contained a ‘+’ sign.</p> <p>It also works when the term in the parentheses contains a ‘-’ sign.</p> <p>In this example, we see that the coefficient <font size="+2" face="Times New Roman">3</font> is distributed over the two terms <font size="+2" face="Times New Roman"><i>x</i></font> and <font size="+2" face="Times New Roman"><i>6</i></font> in the parentheses, giving us the expression <font size="+2" face="Times New Roman">3<i>x</i> - 18</font> which has no parentheses.</p> <p>Here we have a map showing Jose’s position relative to his school.</p> <p>Find a route for him to go to school by clicking on individual street blocks. Click the “Reset” button to start a new route.</p> <p>The total number of blocks for the chosen route are shown in the textbox. Try to find the shortest distance for him to travel.</p> <p>How many different routes give the shortest distance? Click the “Submit” button after entering your answer.</p> <p>There are two routes that give the shortest distance. No matter which of these routes he takes, the distance is three blocks.</p> <p>This example shows us two different properties of numbers... the associative and the commutative properties.</p> <p>The <b>associative property</b> states that the way you group three numbers <font size="+2" face="Times New Roman"><i>a</i></font>, <font size="+2" face="Times New Roman"><i>b</i></font>, and <font size="+2" face="Times New Roman"><i>c</i></font> when you add them, has no effect on their sum.</p> <p>The <b>commutative property</b> states that the order in which you add two numbers <font size="+2" face="Times New Roman"><i>a</i></font> and <font size="+2" face="Times New Roman"><i>b</i></font> has no effect on their sum.</p> <p>Looking at our example, Jose can either go one “<font size="+2" face="Times New Roman">1 + 1</font>” blocks east and then <font size="+2" face="Times New Roman">1</font> block north.</p> <p>Or, he can go <font size="+2" face="Times New Roman">1</font> block north and then “<font size="+2" face="Times New Roman">1 + 1</font>” blocks east. </p> <p>Grouping the eastern blocks at the beginning or the end makes no difference in the total distance. This follows the associative property.</p> <p>Now, we also can look at his possible routes as being in two ‘parts.’ He either can go to school in one <font size="+2" face="Times New Roman">2</font>-block part and then a <font size="+2" face="Times New Roman">1</font>-block part.</p> <p>Or, he can go in a <font size="+2" face="Times New Roman">1</font>-block part, and then a <font size="+2" face="Times New Roman">2</font>-block part.</p> <p>Changing the order of his route doesn’t change the distance. This follows the commutative property.</p> <p>The associative and commutative properties also apply to multiplication.</p> <p>Grouping the numbers differently in this example does not change the product: <font size="+2" face="Times New Roman">(2 • 3)5 = 2(3 • 5)</font></p> <p>Changing the order of the numbers in this other example does not change the product: <font size="+2" face="Times New Roman">6 • 12 = 12 • 6</font></p> <p>Now, here we have a list of equations that are examples of the associative, commutative, and distributive properties.</p> <p>In the box next to each example, enter the number of the property that applies.</p> <p>After entering your answers, click the “Submit” button to show the correct answer next to each box.</p> <p>In future lessons, we will learn how to use these properties to simplify algebraic expressions.</p> <p id="TextCopyright">Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education</p>