The Associative Law of Multiplication
The purpose of this applet is to present an interactive model of multiplication and use that model to describe and explain the associative law of multiplication. The image below shows the initial state of the applet.
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We have a series of wheels at the top of varying sizes arranged from small to large labeled with ratios: “1:1”, “2:1”, “3:1”, etc up to “10:1”. These wheels represent pulley assemblies consisting of a center pulley of size one unit attached to a larger pulley ranging in size from 1 to 10 units. Each ratio is thus the ratio of the size of the outer pulley to the size of the center pulley. Above each of the pulleys is a counter which will keep track of the number of full turns of the pulley. There are some buttons on the right only one of which, the “Reset” button, is initially enabled. The main part of the screen is taken up with an assembly consisting of a “Crank”, consisting of a center pulley of size one unit, six arms used to “turn” it and a counter which tracks the number of full turns of the crank. The crank is connected by a belt to a blue pulley of the same unit size with a counter on the far right that is in turn connected by a belt to a red pulley on the far left. Attached to the belt is a figure representing a wheeled “robot” which rides along a blue line. The blue line is marked into intervals with marks “0”, “1”, “2” and so on. The robot is initially positioned on mark “0”. The crank is turned by placing the mouse cursor over the crank and turning the mouse wheel. (If you don’t have a mouse wheel you can use the arrow keys instead.) One direction of the mouse wheel (the “down” direction) causes the crank to turn to the right. Turning the wheel the opposite direction causes the crank to turn to the left. (For the arrow keys, right arrow or down arrow turn to the right and left arrow or up arrow turn the pulley to the left.) Turning the crank will cause the blue pulley to turn which will cause the belt attached to the robot to turn which will cause the robot to move along the line. Turning the crank to the right causes the robot to move to the right and turning the crank to the left causes the robot to move to the left. Note that the robot is not allowed to move to the left of the “0” mark.
Please experiment by turning the crank to move the robot back and forth. Note that each full turn of the crank to the right turns the blue pulley one turn to the right which moves the robot one mark to the right and each full turn of the crank to the left causes the robot to move one mark to the left (though the robot has to stop at 0!) There is (almost) no limit to how far the robot can move to the right. When the robot reaches the right end of the image, the view will shift to allow us to see more of the line. The figures below show the states of the robot after three and ten turns of the crank. Note that the counter on the blue pulley measures the number of turns which corresponds to the position of the robot.
Position of the Robot after Three turns
Position of the Robot after Ten Turns
Note also that in the initial configuration the number of turns of the crank exactly matches the number of turns of the blue pulley. Next we involve one of the pulleys at the top. Click on one of them, say the “5:1” pulley (which we’ll just call the “five” pulley.) The applet should now appear as follows.
Configuration after clicking on the Five Pulley.
The five pulley has been inserted between the crank and the blue pulley connected by means of two belts. The crank is connected to the center part of the five pulley. Since the crank pulley and the center have the same size, one turn of the crank will result in one turn of the five pulley. The outer part of the five pulley which is five units in size is connected to the blue pulley of size one unit. Because the outer pulley of the five pulley is five times as large as the blue pulley, each full turn of the five pulley will cause the five pulley to turn five times. Thus each turn of the crank will result in one turn of the five pulley which results in five turns of the blue robot pulley which moves the robot five marks. The figure below shows the state after three turns of the crank in this configuration. Please try this yourself.
Configuration after three turns of the Crank with Five Pulley Inserted
Notice that the crank has turned three times which has caused the five pulley to turn three times which has caused the blue pulley to turn 15 = 5 + 5 + 5 times causing the robot to move 15 marks. We are now in a position to model multiplication by compounding. Click on another pulley, say the seven pulley.
Configuration
with the Five Pulley Compounded with the Seven Pulley
(Note that when you add a pulley the robot is placed back in the zero position. You can also do this by pushing the “Reset” button. Pushing “Clear” will remove the pulleys you’ve added so that you can start over.) We now have two connected pulleys between the crank and the blue robot pulley. As before the crank is connected to the center of the five pulley but now the outer part of the five pulley is connected to the center of the seven pulley. Because the five pulley is five times larger than the center pulley, each turn of the five pulley will result in five turns of the seven pulley. Now the outer part of the seven pulley is connected to the blue pulley. Thus each turn of the seven pulley will result in seven turns of the blue pulley. As you can see for yourself, for this system, one turn of the crank results in the following state.
State after One Turn of a “5 x 7” Configuration
We see that the five pulley has turned once which results in five turns of the seven pulley. Each of these turns results in seven turns of the blue pulley for a total of 7 + 7 + 7 + 7 + 7 = 35 turns of the blue pulley moving the robot 35 marks to the right. Thus connecting two pulleys with ratios 5:1 and 7:1 is equivalent to a single pulley with ratio 5 x 7 = 35:1. In general if we compound a pulley with ratio m:1 with a pulley with ratio n:1 then the result will be equivalent to a pulley with ratio (m x n):1 Thus if we correspond numbers to pulleys then multiplication corresponds to compounding. Now of course we need not stop with two pulleys. Insert another, say the three pulley, and turn the crank one time.
State After One Turn of a “5 x 7 x 3” Configuration
In this case one turn of the crank has resulted in one turn of the five pulley which results in five turns of the seven pulley which results in 7+7+7+7+7 = 35 turns of the three pulley which results in
3+3+3+3+3+3+3 + 3+3+3+3+3+3+3 + 3+3+3+3+3+3+3 + 3+3+3+3+3+3+3 + 3+3+3+3+3+3+3 = 105 turns of the blue pulley moving the robot 105 marks. Let’s multiply again, this time by 10. We then have the following state after turning once.
A “5 x 7 x 3 x 10” Configuration
Now let’s get to the associative law. Normally when we think of multiplication we think of it as an operator on two numbers at a time. On the other hand compounding can be applied to as many pulleys as we like. For example, above we compounded four pulleys. If you were doing arithmetic rather than compounding you might interpret “(5 x 7 x 3 )” as “(( 5 x 7) x 3)”, that is first compute (5 x 7) = 35 and then compute 35 x 3 to get 105. We call this way of computing “grouping to the left” because numbers to the left are grouped together before including a number to the right. On the other hand we might also compute by “grouping” to the right. In that case we would interpret “(5 x 7 x 3)” as “(5 x (7 x 3))”. In this case we would first compute (7 x 3) to get 21 and then compute 5 x 21 to get 105 again. We see (35 x 3) = (5 x 21). Why should this be true? Let’s try it for our four number example. We can interpret (5 x 7 x 3 x 10) as
(((5 x 7) x 3) x 10) = ((35 x 3) x 10) = (105 x 10) = 1050. On the other hand we could have also interpreted it as (5 x (7 x (3 x 10)) = (5 x (21 x 10)) = (5 x 210) = 1050. On the other “other” hand we could have also grouped the numbers as ((5 x 7) x (3 x 10)) = (35 x 30) = 1050 and still get the same result. It seems almost magical that these different ways of grouping the numbers in a multiplication should produce the same values. Yet, the fact is that no matter how we group the numbers in multiplying a group of numbers the result will be the same. We might call this fact the “Grouping Law for Multiplication” but traditionally it is called the “Associative Law of Multiplication”. (“Associate” is just another word for “group.”)
Our model can help us understand why the associative law is true. To keep things simple we’ll just show what happens with a 3 pulley chain. Click the “Clear” button so that you can compound a fresh group of pulleys and then compound the five, seven, and three pulleys. Before turning the crank click the “Group Left” button. You should see the following:
Viewing
“(5 x 7 x 3)” as “((5 x 7) x 3)”
All we have done is to draw boxes that indicate how we are viewing the pulleys. The boxes have absolutely no effect on what the pulleys do. In our minds we are thinking of the left two pulleys as being a group which is then connected to the third pulley. Viewing it this way we think of the left two pulleys as forming a pulley equivalent to a 35:1 pulley which is then connected to a 3 pulley resulting in a 105:1 pulley. Now click the “Group Right” button. You should see:
Viewing “(5 x 7 x 3)” as “(5 x (7 x 3))”
In this case, in our minds we have decided to put the 7 and the 3 into their own group and then connect the 5 to that assembly. The “(7x3)” gives a 21:1 assembly and then connecting the 5 pulley to that gives a 105:1 assembly. Try turning the crank once with the group right configuration. The robot will move 105 marks. Now click the reset button to put the Robot back on square 0, click the Group Left button and once again turn the crank. The robot will again move 105 marks. To understand the associative law is to realize that the pulley assembly doesn’t care how we decide to view it! It will behave the same however we decide to draw boxes or even if we decide to draw no boxes. You can get back to the no box picture by clicking “Ungroup”.
Try it yourself. Now try the same experiments with four pulleys or five pulleys. Notice that with four or more pulleys there are more than two ways to group the pulleys. Can you find a way of grouping (8 x 4 x 3 x 5) that’s different from the left and right grouping the applet will show you. How many ways can you find to group the multiplications with 5 numbers, like (8 x 4 x 3 x 5 x 2). Do they all give the same result?