Welcome Math Commanders!

Battle in the Polygon – Color Me Surprised

Alright Math Cadets, let’s use our artistic skills to try and tackle this problem.  We’ve seen a lot of evidence with basic arithmetic, but wouldn’t it be nice if we could explore this problem using a more visual approach? It turns out, we can!

Before we begin, print out the page from the “Materials” tab above. We’ll use it to solve this problem the most fun way, with coloring! On the bottom of the page, show your work as needed. When you are done, be sure to upload a picture or scan of the document.

Fist, we need to rethink the number \(.\bar{9}\). All numbers can be written as a sum, so that’s what we’re going to do. For example, the number 52 can be written as 50+2.  This is actually the fundamental idea of our number system, but we don’t need to get too caught up in that right now. 

We can use the same strategy we used on 52 to write .99 as the sum of two numbers. .99=.9+.09

Can you do the same thing with .9999, except this time it should be four numbers added together?

We can extend this idea to write \(.\bar{9}\) as a sum.  To get \(.\bar{9}\), we don’t need to add 2, or 4 numbers, but instead an INFINITE number of numbers.

Try writing the first 5 parts in a sum (addition) that equals \(.\bar{9}\).

Alright, I promised you something visual, so let’s take what you just did and turn it into a coloring problem!

I’ve sent you a strange looking square (you can download a copy of this from the materials tab above). Let’s see how we can once and for all decide this battle and answer the question, which number is bigger, or are they the same?

When you wrote \(.\bar{9}\) as the sum of an infinite number of numbers, the first five parts should have been .9+.09+.009+.0009+.00009.  Well, As you can see, the square has been broken into 10 parts with vertical lines.  Let’s think of the entire square as 1, so then .9 would mean we color 9 of the 10 parts (.9 is equivalent to 9/10).  Starting on the left side, color in the first 9 bars.  This leaves us with 1/10 left. 

Now, the next part in our sum that equals \(.\bar{9}\) is .09 also known as 9/100.  The bar we have left is only 1/10 of the total, and we want to color 9/100. It turns out, 9/100 is the same thing as 9/10 of 1/10. So, we need to break the last bar into 10 equal parts, and color 9 of them. To make it easy on yourself, make sure the 1 small square you left blank is in the corner.

Alright, now we have just a small square left in the corner. That square represents 1/100 of the entire square. As you can see, it is 1/10 of 1/10, which is 1/100. The next number we need to add is .009, also known as 9/1000. Just like 9/100 was really 9/10 of 1/10, 9/1000 is just 9/10 of 1/100. Since that small square is 1/100 of the entire large square, we again need to break it into 10 parts, then color 9 of them.

Do you see the pattern here?  We’re going to keep taking what’s left, breaking it into 10 pieces, and then coloring 9 of them.  If you keep doing this an INFINITE number of times, will there be anything that doesn’t end up getting colored?

Obviously YOU can’t keep doing this an infinite number of times, so once the uncolored part becomes small enough you can’t break it into 10 pieces, you can stop and upload a picture or scan below.