Is 13×27 EQUAL to 17×23?

Is 13 x 27 equal to 17 x 23?

13 x 27 interprets as 13 groups of 27 or 27 groups of 13.                                                                     17 x 23 interprets as 17 groups of 23 or 23 groups of 17.                                                       Comparing both, we can see that there are less groups of a larger number than groups of a smaller number.  How can we tell if they are the same?

At first glance, most students would say that they ARE the same due to the fact that we’ve just switched the ones places for both numbers. Let’s break these numbers apart and compare.  I am going to use a table to organize them.Screen Shot 2016-04-14 at 9.16.19 AM                  As you can see from the table, the difference is one has 4 groups of 13 and one has 4 groups of 23.  Clearly, the second one is larger.  How much is it larger by?   Further breaking down of the product 4×23 uncovers the difference, which is four groups of ten.

Let’s look at this in a diagram form.

Screen Shot 2016-04-14 at 9.14.03 AM

The black rectangles are the same between the two products, and so are the red ones.  The purple one is the four groups of ten that account for the difference between the two products.

Here is another way to determine which is greater.  Let’s compare the numbers in each of the pair.  

First, 13 and 27, the difference is 14.  That means on a number line, they are 14 numbers apart.  Next, 17 and 23, the difference is 6.  That means on a number line, they are 6 units apart.

So, we found out that 17 x 23  is greater than 13 x 27 and so the closer the numbers are together (smaller difference), the greater their product will be. But why is this so?

I invite you to join me in exploring why this is.  Respond with a comment to this post and I would love to hear from you.

 

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